Lesson video

Hi there.

My name's Ms. Lambow.

You've made such a fantastic choice deciding to join me today to do some maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is Laws of Indices and we'll be looking at raising a power to a power.

This is within the unit Arithmetic Procedures and Index Laws.

By the end of this lesson, you will be able to use the laws of indices to simplify a power raised to another power.

Some key words that we'll be using in today's lesson are exponent, index, coefficient, and power.

A quick recap of what these are.

An exponent is a number positioned above and to the right of a base value.

It indicates repeated multiplication.

An alternative word for this is index and the plural of index is indices.

A numerical coefficient is a constant multiplier of the variables in a term.

And power, 16 is the fourth power of two.

Alternatively, this could be written as two with an exponent of four, which is read as two to the power of four.

Today's lesson is split into three learning cycles.

In the first one, we will look at exploring exponents of exponents.

We will then use the laws of indices.

And in the final learning cycle we will look at changing the base.

Let's get going with that first one.

So exponents of exponents.

Here we've got Alex and Laura.

Question is write seven squared in brackets cubed as a single power of seven.

Alex says, "We have been doing a lot of work recently on exponents and we have found that there are some shortcuts.

' Laura says, "Remember Alex, it's only okay to use shortcuts if you understand why they work." Alex's response is, "Yes Laura, it's fine.

I have understood them.

I was just wondering if there was a shortcut here." Remember, fine to use shortcuts, but you must make sure that you understand why they work.

Do you think there will be a shortcut? Explore what happens when we are finding exponents of exponents.

Sticking with the same question then.

We need to write seven squared cubed as a single power of seven.

Considering order of operations, we can write the bracket in expanded form.

Seven squared is seven multiplied by seven.

What does the exponent mean? It means a repeated multiplication of what is in the bracket three times.

So it's seven multiplied by seven, multiplied by that again, multiplied by that again.

A repeated multiplication of seven squared three times.

Now, we can write this into exponent form.

This is seven to the power of six.

Do you notice anything? Don't worry if you don't.

We're gonna take a look at another one and then hopefully you'll spot something.

Write x the power of four and that's in brackets squared as a single power of x.

Write the bracket in expanded form.

x multiplied by x, multiplied by x, multiplied by x and then we're squaring that.

What does the squared mean though? What does that exponent mean? Yeah, we're going to repeat that multiplication, whatever's in the bracket two times.

We've got the bracket multiplied by that bracket again.

Now we can write that back into exponent form.

It gives us x to the power of eight.

Do you notice anything now? If you notice something before, is it still the same thing? Or if you didn't notice something, do you now? I'm sure you've noticed that what we are doing here is multiplying the exponents.

Four multiplied by two is eight.

Which of the following is the correct response to this? Show that both a the power of four cubed and a cubed the power of four are equal to a the power of 12.

Pause the video.

Decide which of those you think is the correct response and then when you're ready come back.

A is incorrect.

Both of these only show a to the power of 12.

B is also incorrect.

The top line shows a to the power of six squared and the bottom row shows a squared to the power of six.

No surprises then, the final one is correct.

We can see we've got a to the power of four and that's repeated three times, which is a the power of four cubed.

And then we've got a cubed repeated multiplication of that four times.

So that's a cubed to the power of four.

So it was C.

Super quick task now, I'd just like you to have a go at these three questions and then when you're ready, come back.

You can pause the video now.

Well done.

I am not gonna try and read all of those out.

I'm gonna get very tongue tied with all the h's and m's and p's so I'm going to say to you, please pause the video now and then when you've checked your answers come back and we can move on to the second learning cycle for today's lesson.

Well done.

Like I said, we'll now move on to that second learning cycle using the laws of indices.

A generalised form of the power law for exponents is a to the power of m, that's in brackets to the power of n is a to the power of mn.

Some examples.

Four to the power of eight to the power of five is four to the power of 40.

Five to the power of negative three to the power of seven is five to the power of negative 21.

Seven to the power of negative two to the power of negative five is seven to the power of 10 'cause negative two multiplied by negative five is 10.

x squared to the power of negative three is x to the power of negative six.

And then finally, y to the power of three over two to the power of 0.

25 is y to the power of 3/8.

3/2 and we know that 0.

25 is equivalent to 1/4.

When we multiply the fractions we end up with 3/8.

When raising an exponent to a power, we multiply the exponents.

Simplify eight to the power of negative five to the power of three.

Give our answer in index form.

We know that we are going to multiply the exponents.

Negative five multiplied by three is negative 15.

So the answer is eight to the power of negative 15.

Now I'd like you to give this one a go.

Pause the video and then when you've got your answer come back.

And your answer should be two to the power of negative 60.

12 multiplied by negative five is negative 60.

Which of the following is not equivalent to x to the power of 12? Pause the video and decide any of those that are not equivalent to x to the power of 12.

Come back when you've got your answers and we'll check.

What did you decide? A was because negative four multiplied by negative three is 12.

B was not because negative four multiplied by three is negative 12, not 12.

C was because negative two multiplied by negative six is 12.

And D was also equivalent because six multiplied by two equals 12.

The correct answer then was B.

B was the only one that was not equivalent to x the power of 12.

Simplify fully.

In brackets two x squared and then that's cubed.

Alex's answer is eight x to the power of six.

Laura's answer is two x to the power of six.

Whose answer do you agree with? Why do you agree with that answer? Alex's answer is correct.

Let's take a look at why.

We'll start by writing a bracket in expanded form.

Two multiplied by x multiplied by x.

That's the same as two x squared.

It's just written in the expanded form and remember we're cubing that.

Now we're going to write the exponent in expanded form.

We're gonna repeat that multiplication three times.

Using the associative and commutative laws, we can rearrange this expression.

We end up with two multiplied by two multiplied by two, and then a repeated multiplication of x six times.

Then we simplify two multiplied by two multiplied by two is eight and a repeated multiplication of x six times is x to the power of six.

So Alex's answer was right and Laura's answer was incorrect.

What mistake has Laura made? She's forgotten that the exponent of three needs to be applied to everything in the bracket and that's a common mistake.

Lots of people often make that mistake.

Must remember that the two is in the bracket, so therefore that is being cubed.

Also, x squared is being cubed.

So two cubed is eight and x squared cubed is x to the power of six.

Simplify fully two x to the power of negative five and that's in brackets, and then that to the power of four.

Remember, everything in the bracket needs to be raised to the power of four.

Two to the power of four multiplied by x to the power of negative five to the power of four.

Two to power of four is 16.

We're multiplying the exponents so we end up with 16x to the power of negative 20.

Now your turn.

Pauses a video and give this one a go.

Now we can check your answer.

We're gonna cube the coefficient, so we're gonna cube five and we're gonna cube the power of x.

So we end up with 125 then we're multiplying the exponents together.

We end up with 125x to the power of 18.

Is that what you got? Of course you did.

Now we'll taking a look at this one.

A is two squared multiplied by three cubed and multiplied by seven and B is two cubed multiplied by three to the power of five, multiplied by seven squared.

We need to work out the value of A cubed multiplied by B and give our answer in the form two to the power of x, three to the power of y, seven to the power of z where x, y, and z are positive integers.

We'll deal first with A cubed.

So we take A and we cube it.

Remember everything in the bracket needs to be cubed.

Now we can use the index log for powers.

We multiply the exponents, we end up with two to the power of six multiplied by three to the power of nine, multiplied by seven cubed.

We wanted to do a cubed multiplied by B, so we're gonna take A cubed, we've just worked that out and we're gonna multiply that by B, and then we can rearrange the expression.

So I've got my powers of two together, my powers of three together, and my powers of seven together.

And then I can use my the multiplication law for indices, which is to add the exponent.

So we end up with two to the power of nine, multiplied by three to the power of 14, multiplied by seven to the power of five.

Now your turn.

I'd like you to give this one a go please.

Pause the video and then when you're ready, come back.

Great work.

Now let's check that answer.

So, A squared simplified once we've been through all of the steps as two to the power of six, multiplied by three to the power of eight, multiplied by seven to power four.

Then we need to take that and multiply it by B.

So we're gonna rearrange so that the powers of two, powers of three and powers of seven together and then we can apply the multiplication law for indices.

So we're gonna add the exponents.

So two to the power of seven multiplied by three to the power of 10, multiplied by seven to the power of eight.

Remember, two has an exponent of one, we just don't write that exponent of one.

Now for task B.

I'd like you to fully simplify the following.

Pause the video and then when you're ready, come back.

Well done.

And question number two.

Again, pause the video and then come back when you're ready.

Just take real extra care here because I've given you some negative exponents.

Good luck and I'll be here waiting when you get back.

Super work.

Question number one.

A was four to the power of 18.

B, seven to the power of six.

C, x to the power of negative 20.

D, 32a to the power of 15.

E, 64b to the power of negative 14.

and F, 27c to the power of negative three.

I've shown my workings there so you could pause the video if you've made any errors and check that you understand any errors you've made.

And then question two, A was two multiply by three to the power of 11 multiplied by five.

B, two to the power of negative four multiplied by three to the power of 13, multiplied by five to the power of five.

And then finally C, two to the power of nine multiplied by three to the power of four, multiplied by five to the power of negative six.

Now, we can have a look at changing the base.

Simplify three to power of five multiplied by nine squared giving your answer as a single power of three.

Alex says, "We cannot do this as the bases are not the same.

We can only use the multiplication law when the bases are the same." Can you see a way that we could simplify this? Laura says, "Yes Alex, you are right.

But, we can write nine with a base of three." Is that what you decided? And Alex's response is, "Oh yes Laura, you are right we can." Let's take a look at how we're gonna do this then.

Nine can be rewritten as what? Remembering though the base needs to be three as the question asked for our answer as a power of three.

What can we rewrite nine as? Nine is three squared.

We can now rewrite the expression, here's our original expression, but we know that nine is equivalent to three squared, so I'm going to replace the nine with three squared.

Now, we can use the laws of indices.

So we've got three to the power of five multiplied by three squared, squared, which is three to power of five multiplied by three to power of four.

Remember when we're raising to a power, we multiply those exponents.

Two multiplied by two is four.

And then finally we can use the multiplication law, but we can add the exponents giving us three to the power of nine.

Now we're gonna have a go at this one.

Simplify four to the power of five multiplied by eight squared divided by 16 squared.

And we're going to give our answer in index form.

Laura says, "How are we gonna do this? 16 is a power of four but eight is not a power of four." Can you see a way that we could simplify this? Alex says, "But all of them are powers of two." I wonder if that's what you decided.

We need to rewrite four, eight, and 16 as powers of two.

What are each of those written as a power of two? Four is two squared, eight is two cubed, and 16 is two to the power of four.

Substitute four for two squared, we end up with two squared to the power of five.

Eight squared is going to be two cubed squared and then 16 squared is going to be two to the power of four squared.

All I've done here is I have substituted in my powers of two for four, eight, and 16.

We can now use the index law where we're going to multiply the exponents together.

So we end up with two to the power of 10, two to the power of six, and two to the power of eight.

Let's go back to our original equation.

We now know that four to the power of five is two to the power of 10, eight squared was two to the power of six, and 16 squared was two to the power of eight.

And now we can apply our multiplication and division laws for indices.

Therefore, we're gonna add together the first two exponents and subtract the third exponent giving us 10, add six, subtract eight giving a final answer of two to the power of eight.

I'd like you to match each expression to the base that will be used to simplify it.

Pause the video and then we'll check in in a moment to see how you've got on.

So the first one would be five.

25 is five squared, 125 is five cubed.

The second one is four, four cubed is 64, four squared is 16.

The third one is three.

27 is three cubed and nine is three squared.

And then finally the last one was two.

We'll have a go now at simplifying this one.

We'll give our answer in index form.

64 we know is two to the power of six.

32 is two to the power of five.

Now we're going to substitute two to the power of six for 64 and now two to the power of five instead for 32.

We're gonna multiply the exponents giving us two to the power of 18 and two to power of 10.

If we go back to the original question, we were multiplying those two things together and so therefore, the final step is to add those exponents giving us two to the power of 28.

And now your turn.

Pause the video and then when you've got your answer, come back.

This one we were going to write them as powers of five.

25 is five squared, 125 is five cubed.

So we end up with five squared to the power of four, five cubed to the power of three or five cubed, cubed I should say, giving us five to the power of eight and five to the power of nine.

And then this time we were dividing, so we needed to subtract the exponents giving us five to the power of negative one.

Eight subtract nine is negative one.

Now I'd like you please to have a go at task C.

Pause the video and then when you're ready come back and we'll check.

Let's check those answers.

A is two to the power of four.

B, two to the power of six.

C, two to the power of seven.

D, three cubed and E, three to the power of four.

And then you use those to answer question two.

A is two to the power four.

B, three to the power of 17.

And C, two to the power of negative 27.

Obviously if you need to pause the video to look at the workings for those if you've made any errors, which I'm sure you haven't, then you can do that and then check back in with me in a moment.

Summarise what we've been doing during today's lesson.

A generalised form of the power law for exponents is a to the power of m in brackets to the power of n is a to the power of mn.

And those are the examples that we looked at.

So this applies to positive exponents, negative exponents.

It applies to algebraic bases and also the exponent can be decimals or fractions.

When raising an exponent to a power, we multiply the exponents.

Remember to take if there is a coefficient because this must also be raised to the power.

It is possible to simplify, for example, four to power of five, multiplied by eight squared, divided by 16 squared by writing all of the numbers with the same base and then we can apply the power law.

Fantastic work today, well done.

I look forward to seeing you again really soon.

Take care of yourself and goodbye.